The gift of small edits and subtraction

While making the chart on fertility rates (link), I came across a problem that pops up quite often, and is  ignored by most software programs.

Here is an earlier version of the chart I later discarded:

Compare this to the version I published in the blog post:

Aside from adding the chart title, there is one major change. I removed the empty plots from the grid. This is a visualization trick that should be called adding by subtracting. The empty scaffolding on the first chart increases our cognitive load without yielding any benefit. The whitespace brings out the message that only countries in Asia and Africa have fertility rates above 5.0. 

This is a small edit. But small edits accumulate and deliver a big impact. Bear this in mind the next time you make a chart.

 

P.S.

(1) You'd have to use a lower-level coding language to execute this small edit. Most software programs are quite rigid when it comes to making small-multiples (facet) charts.

(2) If there is a next iteration, I'd reverse the Asia and Oceania rows.

 

Visualizing fertility rates around the globe

The following chart dropped on my Twitter feed.

It's an ambitious chart that tries to do a lot. The underlying data set contains fertility rate data from over 200 countries over 20 years.

The basic chart form is a column chart that is curled up into a ball. The column chart is given colors that map to continents. All countries are grouped into five continents. The column chart can only take a single data series, so the 2019 fertility rate is chosen.

Beyond this basic setup, the designer embellishes the chart with a trove of information. Here's a close up:

The first number is the 2019 fertility rate, which means all the data encoded into the columns are also printed on the chart itself. Then, the flag of each country forms the next ring. Then, the name of the country. Finally, in brackets, the percent change in fertility rate between 2000 and 2019.

That is not all. Some contextual information are injected in those arrows that connect the columns to the data labels. A green arrow indicates that the fertility rate is trending lower - which is the case in most countries around the world. Once in a while, a purple arrow pops up. In the above excerpt, Seychelles gets a purple arrow because this island nation has increased the fertility rate from 2000 to 2019.

Also hiding in the background are several dashed rings. I think only the one that partially overlaps with the column chart contains any information - the other rings are inserted for an artistic reason. To decipher this dashed ring, we must look at the inset in the top left corner. We learn that the value of 2.1 children per woman is known as the replacement fertility rate. So it's also possible to assess whether each country is above or below the replacement fertility rate threshold.

[I'm presuming that this replacement threshold is about the births necessary to avoid a population decline. If that's the case, then comparing each country's fertility rate to a global fertility rate threshold is too simplistic because fertility is only one of several key factors driving a country's population growth. A more sophisticated model should generate country-level thresholds.]

***

Data graphics serve many functions. This chart works well as an embellished data table. It does take some time to find a specific country because the columns have been sorted by decreasing 2019 fertility rate but once we locate the column, all the other data fields are clearly laid out.

As a generator of data insights, this chart is less effective. The main insight I obtained from it is a rough ranking of continents, with African countries predominantly having higher fertility rates, followed by Asia and Oceania, then Americas, and finally, Europe which has the lowest fertility rates. If this is the key message, a standard choropleth map brings it out more directly.

***

Here is a small-multiples rendering of the fertility dataset. I chose 1999 values instead of 2000 to make a complete two-decade view.

The columns represent a grouping of countries based on their 1999 fertility rates. The left column contains countries with the lowest number of births per woman, and the fertility rate increases left to right - both within an individual plot and in the grid.

If you're wondering, the hidden vertical axis sorts the countries by their 1999 rank. The lighter colors are 1999 values while the darker colors are 2019 values. For most countries the dots are shifting left over the 20 years. There are some exceptions. I have labeled several of these exceptions (e.g. Kazakhstan and Mongolia), and rendered them in italic.

 

 

 

Charts that ask questions about the German election

In the prior post about Canadian elections, I suggested that designers expand beyond plots of one variable at a time. Today, I look at a project by DataWrapper on the German elections which happened this week. Thanks to long-time blog supporter Antonio for submitting the chart.

The following is the centerpiece of Lisa's work:

CDU/CSU is Angela Merkel's party, represented by the black color. The chart answers one question only: did polls correctly predict election results?

The time period from 1994 to 2021 covers eight consecutive elections (counting the one this week). There are eight vertical blocks on the chart representing each administration. The right vertical edge of each block coincides with an election. The chart is best understood as the superposition of two time series.

You can trace the first time series by following a step function - let your eyes follow the flat lines between elections. This dataset shows the popular vote won by the party at each election, with the value updated after each election. The last vertical block represents an election that has not yet happened when this chart was created. As explained in the footnote, Lisa took the average poll result for the last month leading up to the 2021 election - in the context of this chart, she made the assumption that this cycle of polls will be 100% accurate.

The second time series corresponds to the ragged edges of the gray and black areas. If you ignore the colors, and the flat lines, you'll discover that the ragged edges form a contiguous data series. This line encodes the average popularity of the CDU/CSU party according to election polls.

Thus, the area between the step function and the ragged line measures the gap between polls and election day results. When the polls underestimate the actual outcome, the area is colored gray; when the polls are over-optimistic, the area is colored black. In the last completed election of 2017, Merkel's party underperformed relative to the polls. In fact, the polls in the entire period between the 2013 and 2017 uniformly painted a rosier picture for CDU/CSU than actually happened.

The last vertical block is interpreted a little differently. Since the reference level is the last month of polls (rather than the actual popular vote), the abundance of black indicates that Merkel's party has been suffering from declining poll numbers on the approach of this week's election.

***

The picture shown above seems to indicate that these polls are not particularly good. It appears they have limited ability to self-correct within each election cycle. Aside from the 1998-2002 period, the area colors seldom changed within each cycle. That means if the first polling average overestimated the party's popularity, then all subsequent polling averages were also optimistic. (The original post focused on a single pollster, which exacerbates this issue. Compare the following chart with the above, and you'll find even fewer color changes within cycle here:

Each pollster may be systematically biased but the poll aggregate is less so.)

 

Here's the chart for SDP, which is CDU/CSU's biggest opponent, and likely winner of this week's election:

Overall, this chart has similar features as the CDU/CSU chart. The most recent polls seem to favor the SPD - the pink area indicates that the older polls of this cycle underestimates the last month's poll result.

Both these parties are in long-term decline, with popularity dropping from the 40% range in the 1990s to the 20% range in the 2020s.

One smaller party that seems to have gained followers is the Green party:

The excess of dark green, however, does not augur well for this election.

 

 

 

 

 

Tongue in cheek but a master stroke

Andrew jumped on the Benford bandwagon to do a tongue-in-cheek analysis of numbers in Hollywood movies (link). The key graphic is this:

Benford's Law is frequently invoked to prove (or disprove) fraud with numbers by examining the distribution of first digits. Andrew extracted movies that contain numbers in their names - mostly but not always sequences of movies with sequels. The above histogram (gray columns) are the number of movies with specific first digits. The red line is the expected number if Benford's Law holds. As typical of such analysis, the histogram is closely aligned with the red line, and therefore, he did not find any fraud. 

I'll blog about my reservations about Benford-style analysis on the book blog later - one quick point is: as with any statistical analysis, we should say there is no statistical evidence of fraud (more precisely, of the kind of fraud that can be discovered using Benford's Law), which is different from saying there is no fraud.

***

Andrew also showed a small-multiples chart that breaks up the above chart by movie groups. I excerpted the top left section of the chart below:

The genius in this graphic is easily missed.

Notice that the red lines (which are the expected values if Benford Law holds) appear identical on every single plot. And then notice that the lines don't represent the same values.

It's great to have the red lines look the same everywhere because they represent the immutable Benford reference. Because the number of movies is so small, he's plotting counts instead of proportions. If you let the software decide on the best y-axis range for each plot, the red lines will look different on different charts!

You can find the trick in the R code from Gelman's blog.

First, the maximum value of each plot is set to the total number of observations. Then, the expected Benford proportions are converted into expected Benford counts. The first Benford count is then shown against an axis topping out at the total count, and thus, relatively, what we are seeing are the Benford proportions. Thus, every red line looks the same despite holding different values.

This is a master stroke.

 

 

 

What metaphors give, they take away

Aleks pointed me to the following graphic making the rounds on Twitter:

It's being passed around as an example of great dataviz.

The entire attraction rests on a risque metaphor. The designer is illustrating a claim that Covid-19 causes erectile dysfunction in men.

That's a well-formed question so in using the Trifecta Checkup, that's a pass on the Q corner.

What about the visual metaphor? I advise people to think twice before using metaphors because these devices can give as they can take. This example is no exception. Some readers may pay attention to the orientation but other readers may focus on the size.

I pulled out the tape measure. Here's what I found.

The angle is accurate on the first chart but the diameter has been exaggerated relative to the other. The angle is slightly magnified in the bottom chart which has a smaller circumference.

***

Let's look at the Data to round out our analysis. They come from a study from Italy (link), utilizing survey responses. There were 25 male respondents in the survey who self-reported having had Covid-19. Seven of these submitted answers to a set of five questions that were "suggestive of erectile dysfunction". (This isn't as arbitrary as it sounds - apparently it is an internationally accepted way of conducting reseach.) Seven out of 25 is 28 percent. Because the sample size is small, the 95% confidence range is 10% to 46%.

The researchers then used the propensity scoring method to find 3 matches per each infected person. Each match is a survey respondent who did not self-report having had Covid-19. See this post about a real-world vaccine study to learn more about propensity scoring. Among the 75 non-infected men, 7 were judged to have ED. The 95% range is 3% to 16%.

The difference between the two subgroups is quite large. The paper also includes other research that investigates the mechanisms that can explain the observed correlation. Nevertheless, the two proportions depicted in the chart have wide error bars around them.

I have always had a question about analysis using this type of survey data (including my own work). How do they know that ED follows infection rather than precedes it? One of the inviolable rules of causation is that the effect follows the cause. If it's a series of surveys, the sequencing may be measurable but a single survey presents challenges. 

The headline of the dataviz is "Get your vaccines". This comes from a "story time" moment in the paper. On page 1, under Discussion and conclusion, they inserted the sentence "Universal vaccination against COVID-19 and the personal protective equipment could possibly have the added benefit of preventing sexual dysfunctions." Nothing in the research actually supports this claim. The only time the word "vaccine" appears in the entire paper is on that first page.

"Story time" is the moment in a scientific paper when the researchers - after lulling readers to sleep over some interesting data - roll out statements that are not supported by the data presented before.

***

The graph succeeds in catching people's attention. The visual metaphor works in one sense but not in a different sense.

 

P.S. [8/6/2021] One final note for those who do care about the science: the internet survey not surprisingly has a youth bias. The median age of 25 infected people was 39, maxing out at 45 while the median of the 75 not infected was 42, maxing out at 49.

Ranking data provide context but can also confuse

This dataviz from the Economist had me spending a lot of time clicking around - which means it is a success.

The graphic presents four measures of wellbeing in society - life expectancy, infant mortality rate, murder rate and prison population. The primary goal is to compare nations across those metrics. The focus is on comparing how certain nations (or subgroups) rank against each other, as indicated by the relative vertical position.

The Economist staff has a particular story to tell about racial division in the US. The dotted bars represent the U.S. average. The colored bars are the averages for Hispanic, white and black Americans. The wider the gap between the colored bars, the more variant is the experiences between American races.

The chart shows that the racial gap of life expectancy is the widest. For prison population, the U.S. and its racial subgroups occupy many of the lowest (i.e. least desirable) ranks, with the smallest gap in ranking.

***

The primary element of interactivity is hovering on a bar, which then highlights the four bars corresponding to the particular nation selected. Here is the picture for Thailand:

According to this view of the world, Thailand is a close cousin of the U.S. On each metric, the Thai value clings pretty near the U.S. average and sits within the range by racial groups. I'm surprised to learn that the prison population in Thailand is among the highest in the world.

Unfortunately, this chart form doesn't facilitate comparing Thailand to a country other than the U.S as one can highlight only one country at a time.

***

While the main focus of the chart is on relative comparison through ranking, the reader can extract absolute difference by reading the lengths of the bars.

This is a close-up of the bottom of the prison population metric:

The length of each bar displays the numeric data. The red line is an outlier in this dataset. Black Americans suffer an incarceration rate that is almost three times the national average. Even white Americans (blue line) is imprisoned at a rate higher than most countries around the world.

As noted above, the prison population metric exhibits the smallest gap between racial subgroups. This chart is a great example of why ranking data frequently hide important information. The small gap in ranking masks the extraordinary absolute difference in incareration rates between white and black America.

The difference between rank #1 and rank #2 is enormous.

The opposite situation appears for life expectancy. The life expectancy values are bunched up especially at the top of the scale. The absolute difference between Hispanic and black America is 82 - 75 = 7 years, which looks small because the axis starts at zero. On a ranking scale, Hispanic is roughly in the top 15% while black America is just above the median. The relative difference is huge.

For life expectancy, ranking conveys the view that even a 7-year difference is a big deal because the countries are tightly bunched together. For prison population, ranking shows the view that a multiple fold difference is "unimportant" because a 20-0 blowout and a 10-0 blowout are both heavy defeats.

***

Whenever you transform numeric data to ranks, remember that you are artificially treating the gap between each value and the next value as a constant, even when the underlying numeric gaps show wide variance.

 

 

 

 

 

Further exploration of tessellation density

Last year, I explored using bar-density (and pie-density) charts to illustrate 80/20-type distributions, which are very common in real life (link).

The key advantage of this design is that the most important units (i.e. the biggest stars/creators) are represented by larger pieces while the long tail is shown by little pieces. The skewness is encoded in the density of the tessellation.

So when the following chart showed up on my Twitter feed, I returned to the idea of using tessellation density as a visual cue.

This wbur chart is a good statistical chart - effiicient at communicating the data, but "boring". The only things I'd change is to remove the vertical axis, gridlines, and the decimals.

In concept, the underlying data is similar to the Youtube data. Less than 0.5 percent of Youtubers produced 38% of the views on the platform. The richest 1% of the population took 15% of Harvard's spots; the richest 20% took 70%.

As I explore this further, the analogy falls apart. In the Youtube scenario, the stars should naturally occupy bigger spaces. In the Harvard scenario, letting the children of the top 1% taking up more space on the chart doesn't really make sense since each incoming Harvard student has equal status.

Instead of going down that potential deadend, I investigated how tessellation density can be used for visualization. For one thing, tessellations are pretty things and appealing.

Here is something I created:

The chart is read vertically by comparing Harvard's selection of students with the hypothetical "ideal" of equal selection. (I don't agree that this type of equality is the right thing but let me focus on the visualization here.) This, selectivity is coded in the density. Selectivity is defined here as the over/under representation. Harvard is more "selective" in lower-income groups.

In the first and second columns, we see that Harvard's densities are lower than the densities as expected in the general population, indicating that the poorest 20%, and the middle 20% of the population are under-represented in Harvard's student body. Then in the third column, the comparison flips. The density in the top box is about 3-4 times as high as the bottom box. You may have to expand the graphic to see the 1% slither, which also shows a much higher density in the top box.

I was surprised by how well I was able to eyeball the relative densities. You can try it and let me know how you fare.

(There is even a trick to do this. From the diagram with larger pieces, pick a representative piece. Then, roughly estimate how many smaller pieces from the other tessellation can fit into that representative piece. Using this guideline, I estimate that the ratios of the densities to be 1:6, 1:2, 3:1, 10:1. The actual ratios are 1:6.7, 1:2.5, 3:1, 15:1. I find that my intuition gets me most of the way there even if I don't use this trick.)

Density encoding is under-used as a visual cue. I think our ability to compare densities is surprisingly good (when the units are not overlapping). Of course, you wouldn't use density if you need to be precise, just as you wouldn't use color, or circular areas. Nevertheless, there are many occasions where you can afford to be less precise, and you'd like to spice up your charts.

Plotting the signal or the noise

Antonio alerted me to the following graphic that appeared in the Economist. This is a playful (?) attempt to draw attention to racism in the game of football (soccer).

The analyst proposed that non-white players have played better in stadiums without fans due to Covid19 in 2020 because they have not been distracted by racist abuse from fans, using Italy's Serie A as the case study.

The chart struggles to bring out this finding. There are many lines that criss-cross. The conclusion is primarily based on the two thick lines - which show the average performance with and without fans of white and non-white players. The blue line (non-white) inched to the right (better performance) while the red line (white) shifted slightly to the left.

If the reader wants to understand the chart fully, there's a lot to take in. All (presumably) players are ranked by the performance score from lowest to highest into ten equally sized tiers (known as "deciles"). They are sorted by the 2019 performance when fans were in the stadiums. Each tier is represented by the average performance score of its members. These are the values shown on the top axis labeled "with fans".

Then, with the tiers fixed, the players are rated in 2020 when stadiums were empty. For each tier, an average 2020 performance score is computed, and compared to the 2019 performance score.

The following chart reveals the structure of the data:

The players are lined up from left to right, from the worst performers to the best. Each decile is one tenth of the players, and is represented by the average score within the tier. The vertical axis is the actual score while the horizontal axis is a relative ranking - so we expect a positive correlation.

The blue line shows the 2019 (with fans) data, which are used to determine tier membership. The gray dotted line is the 2020 (no fans) data - because they don't decide the ranking, it's possible that the average score of a lower tier (e.g. tier 3 for non-whites) is higher than the average score of a higher tier (e.g. tier 4 for non-whites).

What do we learn from the graphic?

It's very hard to know if the blue and gray lines are different by chance or by whether fans were in the stadium. The maximum gap between the lines is not quite 0.2 on the raw score scale, which is roughly a one-decile shift. It'd be interesting to know the variability of the score of a given player across say 5 seasons prior to 2019. I suspect it could be more than 0.2. In any case, the tiny shifts in the averages (around 0.05) can't be distinguished from noise.

***

This type of analysis is tough to do. Like other observational studies, there are multiple problems of biases and confounding. Fan attendance was not the only thing that changed between 2019 and 2020. The score used to rank players is a "Fantacalcio algorithmic match-level fantasy-football score." It's odd that real-life players should be judged by their fantasy scores rather than their on-the-field performance.

The causal model appears to assume that every non-white player gets racially abused. At least, the analyst didn't look at the curves above and conclude, post-hoc, that players in the third decile are most affected by racial abuse - which is exactly what has happened with the observational studies I have featured on the book blog recently.

Being a Serie A fan, I happen to know non-white players are a small minority so the error bars are wider, which is another issue to think about. I wonder if this factor by itself explains the shifts in those curves. The curve for white players has a much higher sample size thus season-to-season fluctuations are much smaller (regardless of fans or no fans).

 

 

 

 

Illustrating differential growth rates

Reader Mirko was concerned about a video published in Germany that shows why the new coronavirus variant is dangerous. He helpfully provided a summary of the transcript:

The South African and the British mutations of the SARS-COV-2 virus are spreading faster than the original virus. On average, one infected person infects more people than before. Researchers believe the new variant is 50 to 70 % more transmissible.

Here are two key moments in the video:

This seems to be saying the original virus (left side) replicates 3 times inside the infected person while the new variant (right side) replicates 19 times. So we have a roughly 6-fold jump in viral replication.

Later in the video, it appears that every replicate of the old virus finds a new victim while the 19 replicates of the new variant land on 13 new people, meaning 6 replicates didn't find a host.

As Mirko pointed out, the visual appears to have run away from the data. (In our Trifecta Checkup, we have a problem with the arrow between the D and the V corners. What the visual is saying is not aligned with what the data are saying.)

***

It turns out that the scientists have been very confusing when talking about the infectiousness of this new variant. The most quoted line is that the British variant is "50 to 70 percent more transmissible". At first, I thought this is a comment on the famous "R number". Since the R number around December was roughly 1 in the U.K, the new variant might bring the R number up to 1.7.

However, that is not the case. From this article, it appears that being 5o to 70 percent more transmissible means R goes up from 1 to 1.4. R is interpreted as the average number of people infected by one infected person.

Mirko wonders if there is a better way to illustrate this. I'm sure there are many better ways. Here's one I whipped up:

The left side is for the 40% higher R number. Both sides start at the center with 10 infected people. At each time step, if R=1 (right side), each of the 10 people infects 10 others, so the total infections increase by 10 per time step. It's immediately obvious that a 40% higher R is very serious indeed. Starting with 10 infected people, in 10 steps, the total number of infections is almost 1,000, almost 10 times higher than when R is 1.

The lines of the graphs simulate the transmission chains. These are "average" transmission chains since R is an average number.

 

P.S. [1/29/2021: Added the missing link to the article in which it is reported that 50-70 percent more transmissible implies R increasing by 40%.]

 

 

Atypical time order and bubble labeling

This chart appeared in a Charles Schwab magazine in Summer, 2019.

This bubble chart does not print any data labels. The bubbles take our attention but the designer realizes that the actual values of the volatility are not intuitive numbers. The same is true of any standard deviation numbers. If you're told SD of a data series is 3, it doesn't tell you much by itself.

I first transformed this chart into the equivalent column chart:

Two problems surface on the axes.

For the time axis, the years are jumbled. Readers experience vertigo, as we try to figure out how to read the chart. Our expectation that time moves left to right is thwarted. This ordering also requires every single year label to be present.

For the vertical axis, I could have left out the numbers completely. They are not really meaningful. These represent the areas of the bubbles but only relative to how I measured them.

***

In the next version, I sorted time in the conventional manner. Following Tufte's classic advice, only the tops of the columns are plotted.

What you see is that this ordering is much easier to comprehend. Figuring out that 2018 is an average year in terms of volatility is not any harder than in the original. In fact, we can reproduce the order of the previous chart just by letting our eyes sweep top to bottom.

To make it even easier to read the vertical axis, I converted the numbers into an index, with the average volatility as 100 %  (assigned to 0% on the chart) .

Now, you can see that 2018 is roughly at the average while 2008 is 400% above the average level. (How should we interpret this statement? That's a question I pose to my statistics students. It's not intuitive how one should interpret the statement that the standard deviation is 5 times higher.)